Finite Group Databasebeta

← #530402

Subgroups — #530402

Order 512.

2706 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2332C212yesmaximal
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#14685C211yesmaximal, characteristic
4#14514C214yesmaximal
5#14501C212yesmaximal, complemented
6#14487C212yesmaximal, complemented
7#14297C212yesmaximal
8#14113C214yesmaximal, complemented
9#13926C212yesmaximal, complemented
10#13884C212yesmaximal, complemented
11#13884C212yesmaximal, complemented
12#13876C212yesmaximal
13#13876C212yesmaximal
14C4⋊C42C212yesmaximal, complemented
15C4×#380C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2332—216no—
20C4×C22⋊Q8C4116yescomplemented
21C2×#391C2214yes—
22C2×#391C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#388C2214yes—
25C2×#388C2214yes—
26C2×#387C2214yes—
27C2×#387C2214yes—
28C2×#387C2212yes—
29C2×#385C2212yes—
30C2×#384C2212yes—
31C2×#384C2212yes—
32C2×#381C2212yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C2214yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71#1434C2212yes—
72#1434C2212yes—
73#1434C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C4×#95C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×C22⋊Q8C2×C418yes—
82C2×C22⋊Q8D418yes—
83C2×C22⋊Q8Q818yes—
84C2×C4×Q8—42no—
85C2×C4×Q8—42no—
86C2×C4×Q8C2×C418yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C42—42no—
107C22×C42—42no—
108C22×C42C2×C418yes—
109C22×C42C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#391—216no—
120#391C2×C418yes—
121#389C2×C418yes—
122#388—216no—
123#388—216no—
124#387—216no—
125#387—216no—
126#387C2×C4116yes—
127#385C2×C4116yes—
128#384—28no—
129#384—44no—
130#381—28no—
131#380—28no—
132#380—42no—
133#380—42no—
134#380C2×C418yes—
135C4×C4⋊C4—216no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—44no—
138C4×C4⋊C4—42no—
139C4×C4⋊C4—42no—
140C4×C4⋊C4C2×C4116yes—
141C4×C4⋊C4C2×C418yes—
142C4×C22⋊C4—28no—
143C4×C22⋊C4C2×C4116yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C25C2411yescharacteristic
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8C22×C414yes—
176C22×Q8C2×D414yes—
177C22×Q8C2×Q814yes—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—41no—
183C23×C4—41no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4C22×C414yes—
191C23×C4C22×C414yes—
192C23×C4C22×C414yes—
193C23×C4C2×D414yes—
194C23×C4C2×Q814yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C22⋊Q8C4⋊C4164yescomplemented
210C4×Q8—416no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C418yes—
247C2×C4⋊C4C22×C414yes—
248C2×C4⋊C4C2×D418yes—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×Q814yes—
251C2×C4⋊C4C2×Q818yes—
252C2×C4⋊C4#4118yes—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C2×D418yes—
267C2×C22⋊C4C2×Q818yes—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—28no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—416no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42—41no—
285C2×C42—41no—
286C2×C42—44no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—44no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42C22×C418yes—
294C2×C42C22×C414yes—
295C2×C42C22×C418yes—
296C2×C42C22×C414yes—
297C2×C42C22×C414yes—
298C2×C42C22×C414yes—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—42no—
316#95—42no—
317#95—44no—
318#95—42no—
319#95—42no—
320#95C22×C418yes—
321#95C22×C418yes—
322#95C22×C418yes—
323#95C22×C418yes—
324#95C22×C418yes—
325C24—22no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—21no—
333C24C23×C412yes—
334C24C22×D412yes—
335C24C22×Q812yes—
336C24C2511yescharacteristic
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8C2×C4⋊C4116yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4C2×C4⋊C4116yes—
428C22×C4C23×C412yes—
429C22×C4C23×C414yes—
430C22×C4C23×C412yes—
431C22×C4C23×C412yes—
432C22×C4C23×C414yes—
433C22×C4C23×C412yes—
434C22×C4C23×C414yes—
435C22×C4C23×C414yes—
436C22×C4C23×C412yes—
437C22×C4C23×C412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×Q814yes—
443C22×C4C22×Q814yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C2×#4118yes—
447C22×C4C2×#4114yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+414yes—
451C4⋊C4—216no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C4⋊C4C2×C4⋊C4116yes—
464C4⋊C4C2×C4⋊C4132yes—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C22⋊C4C2×C4⋊C4132yes—
470C42—216no—
471C42—28no—
472C42—28no—
473C42—216no—
474C42—44no—
475C42—44no—
476C42—416no—
477C42—84no—
478C42—84no—
479C42—88no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23C22×C4⋊C414yes—
503C23C24×C411yescharacteristic
504C23C23×D411yescharacteristic
505C23C23×Q811yescharacteristic
506C23C22×#4112yes—
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×2−1+412yes—
510C23#58512yes—
511C23#58511yescharacteristic
512C23#58511yescharacteristic
513C23#58512yes—
514Q8—416no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—416no—
558C2×C4—416no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—416no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4C22×C4⋊C418yes—
600C2×C4C22×C4⋊C418yes—
601C2×C4C22×C4⋊C414yes—
602C2×C4C22×C4⋊C414yes—
603C2×C4C4×#4114yes—
604C2×C4#51814yes—
605C2×C4#51914yes—
606C2×C4#53918yes—
607C2×C4#54218yes—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22C23×C4⋊C411yescharacteristic
620C22C2×C4×#4111yescharacteristic
621C22C2×#51811yescharacteristic
622C22C2×#51911yescharacteristic
623C22#342811yescharacteristic
624C22#342811yescharacteristic
625C22#342811yescharacteristic
626C22C2×#53912yes—
627C22C2×#54212yes—
628C22D4×#4112yes—
629C22Q8×#4112yes—
630C22#349611yescharacteristic
631C22#349611yescharacteristic
632C22#349712yes—
633C22#350111yescharacteristic
634C22#350111yescharacteristic
635C22#350212yes—
636C22#352112yes—
637C22#352412yes—
638C22#353912yes—
639C22#354012yes—
640C22#355712yes—
641C22#355712yes—
642C4—28no—
643C4—24no—
644C4—24no—
645C4—28no—
646C4—42no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—44no—
651C4—48no—
652C4—84no—
653C4—84no—
654C4—84no—
655C4#233218yescomplemented
656C2—44no—
657C2C2×#233211yescharacteristic
658C2C4⋊C4×#4111yescharacteristic
659C2#3418111yescharacteristic
660C2#3418211yescharacteristic
661C2#3424411yescharacteristic
662C2#3436111yescharacteristic
663C2#3437511yescharacteristic
664C2#3940912yes—
665C2#3941112yes—
666C2#3966512yes—
667C2#3966712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #14501C2semidirect
6. #14487C2semidirect
8. #14113C2semidirect
9. #13926C2semidirect
10. #13884C2semidirect
11. #13884C2semidirect
14. C4⋊C42C2semidirect
20. C4×C22⋊Q8C4semidirect
209. C22⋊Q8C4⋊C4semidirect
655. C4#2332semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#23322
C2×C4×C22⋊Q82
#146852
#145142
#145012
#144872
#142972
#141132
#139262
#138842
#138762
C4⋊C422
C4×#3802